Question

calculating differences of complex numbers in po
consider the numbers $z = \sqrt{2}(\cos 45^{\circ} + i\sin 45^{\circ})$ and $w = 2(\cos 90^{\circ} + i\sin 90^{\circ})$.
fill in the blanks to complete the statements.
in rectangular form, $z - w = 1 - \square i$
in polar form $z - w = \sqrt{2}(\cos \square^{\circ} + i\sin \square^{\circ})$

Explanation:

Step1: Convert \( z \) to rectangular form

We know that for a complex number in polar form \( r(\cos\theta + i\sin\theta) \), the rectangular form is \( r\cos\theta + ir\sin\theta \). For \( z = \sqrt{2}(\cos45^{\circ}+i\sin45^{\circ}) \), we calculate the real and imaginary parts:
\(\cos45^{\circ}=\sin45^{\circ}=\frac{\sqrt{2}}{2}\)
So, the real part of \( z \) is \( \sqrt{2}\times\frac{\sqrt{2}}{2}=\frac{2}{2} = 1 \)
The imaginary part of \( z \) is \( \sqrt{2}\times\frac{\sqrt{2}}{2}=1 \)
Thus, \( z = 1 + i \)

Step2: Convert \( w \) to rectangular form

For \( w = 2(\cos90^{\circ}+i\sin90^{\circ}) \), we know that \( \cos90^{\circ}=0 \) and \( \sin90^{\circ}=1 \)
So, the real part of \( w \) is \( 2\times0 = 0 \)
The imaginary part of \( w \) is \( 2\times1 = 2 \)
Thus, \( w = 0 + 2i = 2i \)

Step3: Calculate \( z - w \) in rectangular form

\( z - w=(1 + i)-2i = 1 - i \)
So, the blank in \( z - w = 1-\square i \) is \( 1 \)

Step4: Convert \( z - w = 1 - i \) to polar form

First, find the modulus \( r \):
\( r=\sqrt{1^{2}+(-1)^{2}}=\sqrt{1 + 1}=\sqrt{2} \)
Then, find the argument \( \theta \). Since the real part is positive and the imaginary part is negative, the complex number is in the fourth quadrant.
\( \tan\theta=\frac{-1}{1}=-1 \)
So, \( \theta = 360^{\circ}-45^{\circ}=315^{\circ} \) (or we can also use \( - 45^{\circ} \), but in the range of \( 0^{\circ}\) to \( 360^{\circ}\), it is \( 315^{\circ} \))
Thus, in polar form \( z - w=\sqrt{2}(\cos315^{\circ}+i\sin315^{\circ}) \)

Answer:

For the first blank (in \( z - w = 1-\square i \)): \( 1 \)

For the second and third blanks (in \( z - w=\sqrt{2}(\cos\square^{\circ}+i\sin\square^{\circ}) \)): \( 315 \), \( 315 \)